Polar-coordinate Laplacian identity
= Polar-coordinate Laplacian identity
{title2=$\widetilde\Delta=-\partial_r^2-\frac d r\partial_r+r^{-2}\Delta_{S^d}$}
In $\mathbb R^{d+1}$, $g=dr^2+r^2g_{S^d}$ and the volume density is $r^d$. With the nonnegative <Laplacian> convention, $\widetilde\Delta=-\partial_r^2-(d/r)\partial_r+r^{-2}\Delta_{S^d}$. The identity separates radial homogeneity from the angular <Laplace-Beltrami operator> and derives the <spectrum of the Laplacian on a sphere>.